Integrand size = 133, antiderivative size = 31 \[ \int \frac {e^{-\frac {2 e^x}{x}} \left (-1250 x+e^x (-1250+1250 x)+\left (-1250 x+e^x (-2500+2500 x)\right ) \log (x)+e^x (-1250+1250 x) \log ^2(x)+e^{\frac {2 e^x}{x}} \left (-200 x+2 x^2-200 x \log (x)\right )+e^{\frac {e^x}{x}} \left (e^x (500-500 x)+1000 x+\left (e^x (1000-1000 x)+1000 x\right ) \log (x)+e^x (500-500 x) \log ^2(x)\right )\right )}{x^2} \, dx=2+2 x-\left (-2+5 e^{-\frac {e^x}{x}}\right )^2 (5+5 \log (x))^2 \]
Time = 0.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {e^{-\frac {2 e^x}{x}} \left (-1250 x+e^x (-1250+1250 x)+\left (-1250 x+e^x (-2500+2500 x)\right ) \log (x)+e^x (-1250+1250 x) \log ^2(x)+e^{\frac {2 e^x}{x}} \left (-200 x+2 x^2-200 x \log (x)\right )+e^{\frac {e^x}{x}} \left (e^x (500-500 x)+1000 x+\left (e^x (1000-1000 x)+1000 x\right ) \log (x)+e^x (500-500 x) \log ^2(x)\right )\right )}{x^2} \, dx=2 \left (x-100 \log (x)-50 \log ^2(x)-\frac {625}{2} e^{-\frac {2 e^x}{x}} (1+\log (x))^2+250 e^{-\frac {e^x}{x}} (1+\log (x))^2\right ) \]
Integrate[(-1250*x + E^x*(-1250 + 1250*x) + (-1250*x + E^x*(-2500 + 2500*x ))*Log[x] + E^x*(-1250 + 1250*x)*Log[x]^2 + E^((2*E^x)/x)*(-200*x + 2*x^2 - 200*x*Log[x]) + E^(E^x/x)*(E^x*(500 - 500*x) + 1000*x + (E^x*(1000 - 100 0*x) + 1000*x)*Log[x] + E^x*(500 - 500*x)*Log[x]^2))/(E^((2*E^x)/x)*x^2),x ]
2*(x - 100*Log[x] - 50*Log[x]^2 - (625*(1 + Log[x])^2)/(2*E^((2*E^x)/x)) + (250*(1 + Log[x])^2)/E^(E^x/x))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{-\frac {2 e^x}{x}} \left (e^{\frac {2 e^x}{x}} \left (2 x^2-200 x-200 x \log (x)\right )-1250 x+e^x (1250 x-1250)+e^x (1250 x-1250) \log ^2(x)+e^{\frac {e^x}{x}} \left (e^x (500-500 x)+1000 x+e^x (500-500 x) \log ^2(x)+\left (e^x (1000-1000 x)+1000 x\right ) \log (x)\right )+\left (e^x (2500 x-2500)-1250 x\right ) \log (x)\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^{-\frac {2 e^x}{x}} \left (e^{\frac {2 e^x}{x}} x+500 e^{\frac {e^x}{x}}-100 e^{\frac {2 e^x}{x}}+500 e^{\frac {e^x}{x}} \log (x)-100 e^{\frac {2 e^x}{x}} \log (x)-625 \log (x)-625\right )}{x}-\frac {250 e^{x-\frac {2 e^x}{x}} \left (2 e^{\frac {e^x}{x}}-5\right ) (x-1) (\log (x)+1)^2}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -1250 \int \frac {e^{x-\frac {2 e^x}{x}}}{x^2}dx+500 \int \frac {e^{x-\frac {e^x}{x}}}{x^2}dx+2500 \int \frac {\int \frac {e^{x-\frac {2 e^x}{x}}}{x^2}dx}{x}dx-1000 \int \frac {\int \frac {e^{x-\frac {e^x}{x}}}{x^2}dx}{x}dx-1250 \int \frac {e^{x-\frac {2 e^x}{x}} \log ^2(x)}{x^2}dx+500 \int \frac {e^{x-\frac {e^x}{x}} \log ^2(x)}{x^2}dx-2500 \log (x) \int \frac {e^{x-\frac {2 e^x}{x}}}{x^2}dx+1000 \log (x) \int \frac {e^{x-\frac {e^x}{x}}}{x^2}dx-1250 \int \frac {e^{-\frac {2 e^x}{x}}}{x}dx+1000 \int \frac {e^{-\frac {e^x}{x}}}{x}dx+1250 \int \frac {e^{x-\frac {2 e^x}{x}}}{x}dx-500 \int \frac {e^{x-\frac {e^x}{x}}}{x}dx+1250 \int \frac {\int \frac {e^{-\frac {2 e^x}{x}}}{x}dx}{x}dx-1000 \int \frac {\int \frac {e^{-\frac {e^x}{x}}}{x}dx}{x}dx-2500 \int \frac {\int \frac {e^{x-\frac {2 e^x}{x}}}{x}dx}{x}dx+1000 \int \frac {\int \frac {e^{x-\frac {e^x}{x}}}{x}dx}{x}dx+1250 \int \frac {e^{x-\frac {2 e^x}{x}} \log ^2(x)}{x}dx-500 \int \frac {e^{x-\frac {e^x}{x}} \log ^2(x)}{x}dx-1250 \log (x) \int \frac {e^{-\frac {2 e^x}{x}}}{x}dx+1000 \log (x) \int \frac {e^{-\frac {e^x}{x}}}{x}dx+2500 \log (x) \int \frac {e^{x-\frac {2 e^x}{x}}}{x}dx-1000 \log (x) \int \frac {e^{x-\frac {e^x}{x}}}{x}dx+2 x-100 \log ^2(x)-200 \log (x)\) |
Int[(-1250*x + E^x*(-1250 + 1250*x) + (-1250*x + E^x*(-2500 + 2500*x))*Log [x] + E^x*(-1250 + 1250*x)*Log[x]^2 + E^((2*E^x)/x)*(-200*x + 2*x^2 - 200* x*Log[x]) + E^(E^x/x)*(E^x*(500 - 500*x) + 1000*x + (E^x*(1000 - 1000*x) + 1000*x)*Log[x] + E^x*(500 - 500*x)*Log[x]^2))/(E^((2*E^x)/x)*x^2),x]
3.30.46.3.1 Defintions of rubi rules used
Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84
| method | result | size |
| risch | \(-100 \ln \left (x \right )^{2}+2 x -200 \ln \left (x \right )+\left (500 \ln \left (x \right )^{2}+1000 \ln \left (x \right )+500\right ) {\mathrm e}^{-\frac {{\mathrm e}^{x}}{x}}+\left (-625 \ln \left (x \right )^{2}-1250 \ln \left (x \right )-625\right ) {\mathrm e}^{-\frac {2 \,{\mathrm e}^{x}}{x}}\) | \(57\) |
| parallelrisch | \(-\frac {\left (100 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{x}}{x}} \ln \left (x \right )^{2} x +200 \ln \left (x \right ) {\mathrm e}^{\frac {2 \,{\mathrm e}^{x}}{x}} x -2 \,{\mathrm e}^{\frac {2 \,{\mathrm e}^{x}}{x}} x^{2}-500 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{x}} \ln \left (x \right )^{2} x +625 x \ln \left (x \right )^{2}-1000 \ln \left (x \right ) {\mathrm e}^{\frac {{\mathrm e}^{x}}{x}} x +1250 x \ln \left (x \right )-500 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{x}} x +625 x \right ) {\mathrm e}^{-\frac {2 \,{\mathrm e}^{x}}{x}}}{x}\) | \(111\) |
int(((-200*x*ln(x)+2*x^2-200*x)*exp(exp(x)/x)^2+((-500*x+500)*exp(x)*ln(x) ^2+((-1000*x+1000)*exp(x)+1000*x)*ln(x)+(-500*x+500)*exp(x)+1000*x)*exp(ex p(x)/x)+(1250*x-1250)*exp(x)*ln(x)^2+((2500*x-2500)*exp(x)-1250*x)*ln(x)+( 1250*x-1250)*exp(x)-1250*x)/x^2/exp(exp(x)/x)^2,x,method=_RETURNVERBOSE)
-100*ln(x)^2+2*x-200*ln(x)+(500*ln(x)^2+1000*ln(x)+500)*exp(-exp(x)/x)+(-6 25*ln(x)^2-1250*ln(x)-625)*exp(-2*exp(x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (27) = 54\).
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {e^{-\frac {2 e^x}{x}} \left (-1250 x+e^x (-1250+1250 x)+\left (-1250 x+e^x (-2500+2500 x)\right ) \log (x)+e^x (-1250+1250 x) \log ^2(x)+e^{\frac {2 e^x}{x}} \left (-200 x+2 x^2-200 x \log (x)\right )+e^{\frac {e^x}{x}} \left (e^x (500-500 x)+1000 x+\left (e^x (1000-1000 x)+1000 x\right ) \log (x)+e^x (500-500 x) \log ^2(x)\right )\right )}{x^2} \, dx=-{\left (2 \, {\left (50 \, \log \left (x\right )^{2} - x + 100 \, \log \left (x\right )\right )} e^{\left (\frac {2 \, e^{x}}{x}\right )} - 500 \, {\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} e^{\left (\frac {e^{x}}{x}\right )} + 625 \, \log \left (x\right )^{2} + 1250 \, \log \left (x\right ) + 625\right )} e^{\left (-\frac {2 \, e^{x}}{x}\right )} \]
integrate(((-200*x*log(x)+2*x^2-200*x)*exp(exp(x)/x)^2+((-500*x+500)*exp(x )*log(x)^2+((-1000*x+1000)*exp(x)+1000*x)*log(x)+(-500*x+500)*exp(x)+1000* x)*exp(exp(x)/x)+(1250*x-1250)*exp(x)*log(x)^2+((2500*x-2500)*exp(x)-1250* x)*log(x)+(1250*x-1250)*exp(x)-1250*x)/x^2/exp(exp(x)/x)^2,x, algorithm=\
-(2*(50*log(x)^2 - x + 100*log(x))*e^(2*e^x/x) - 500*(log(x)^2 + 2*log(x) + 1)*e^(e^x/x) + 625*log(x)^2 + 1250*log(x) + 625)*e^(-2*e^x/x)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).
Time = 14.51 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int \frac {e^{-\frac {2 e^x}{x}} \left (-1250 x+e^x (-1250+1250 x)+\left (-1250 x+e^x (-2500+2500 x)\right ) \log (x)+e^x (-1250+1250 x) \log ^2(x)+e^{\frac {2 e^x}{x}} \left (-200 x+2 x^2-200 x \log (x)\right )+e^{\frac {e^x}{x}} \left (e^x (500-500 x)+1000 x+\left (e^x (1000-1000 x)+1000 x\right ) \log (x)+e^x (500-500 x) \log ^2(x)\right )\right )}{x^2} \, dx=2 x + \left (- 625 \log {\left (x \right )}^{2} - 1250 \log {\left (x \right )} - 625\right ) e^{- \frac {2 e^{x}}{x}} + \left (500 \log {\left (x \right )}^{2} + 1000 \log {\left (x \right )} + 500\right ) e^{- \frac {e^{x}}{x}} - 100 \log {\left (x \right )}^{2} - 200 \log {\left (x \right )} \]
integrate(((-200*x*ln(x)+2*x**2-200*x)*exp(exp(x)/x)**2+((-500*x+500)*exp( x)*ln(x)**2+((-1000*x+1000)*exp(x)+1000*x)*ln(x)+(-500*x+500)*exp(x)+1000* x)*exp(exp(x)/x)+(1250*x-1250)*exp(x)*ln(x)**2+((2500*x-2500)*exp(x)-1250* x)*ln(x)+(1250*x-1250)*exp(x)-1250*x)/x**2/exp(exp(x)/x)**2,x)
2*x + (-625*log(x)**2 - 1250*log(x) - 625)*exp(-2*exp(x)/x) + (500*log(x)* *2 + 1000*log(x) + 500)*exp(-exp(x)/x) - 100*log(x)**2 - 200*log(x)
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {e^{-\frac {2 e^x}{x}} \left (-1250 x+e^x (-1250+1250 x)+\left (-1250 x+e^x (-2500+2500 x)\right ) \log (x)+e^x (-1250+1250 x) \log ^2(x)+e^{\frac {2 e^x}{x}} \left (-200 x+2 x^2-200 x \log (x)\right )+e^{\frac {e^x}{x}} \left (e^x (500-500 x)+1000 x+\left (e^x (1000-1000 x)+1000 x\right ) \log (x)+e^x (500-500 x) \log ^2(x)\right )\right )}{x^2} \, dx=500 \, {\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} e^{\left (-\frac {e^{x}}{x}\right )} - 625 \, {\left (\log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} e^{\left (-\frac {2 \, e^{x}}{x}\right )} - 100 \, \log \left (x\right )^{2} + 2 \, x - 200 \, \log \left (x\right ) \]
integrate(((-200*x*log(x)+2*x^2-200*x)*exp(exp(x)/x)^2+((-500*x+500)*exp(x )*log(x)^2+((-1000*x+1000)*exp(x)+1000*x)*log(x)+(-500*x+500)*exp(x)+1000* x)*exp(exp(x)/x)+(1250*x-1250)*exp(x)*log(x)^2+((2500*x-2500)*exp(x)-1250* x)*log(x)+(1250*x-1250)*exp(x)-1250*x)/x^2/exp(exp(x)/x)^2,x, algorithm=\
500*(log(x)^2 + 2*log(x) + 1)*e^(-e^x/x) - 625*(log(x)^2 + 2*log(x) + 1)*e ^(-2*e^x/x) - 100*log(x)^2 + 2*x - 200*log(x)
\[ \int \frac {e^{-\frac {2 e^x}{x}} \left (-1250 x+e^x (-1250+1250 x)+\left (-1250 x+e^x (-2500+2500 x)\right ) \log (x)+e^x (-1250+1250 x) \log ^2(x)+e^{\frac {2 e^x}{x}} \left (-200 x+2 x^2-200 x \log (x)\right )+e^{\frac {e^x}{x}} \left (e^x (500-500 x)+1000 x+\left (e^x (1000-1000 x)+1000 x\right ) \log (x)+e^x (500-500 x) \log ^2(x)\right )\right )}{x^2} \, dx=\int { \frac {2 \, {\left (625 \, {\left (x - 1\right )} e^{x} \log \left (x\right )^{2} + 625 \, {\left (x - 1\right )} e^{x} + {\left (x^{2} - 100 \, x \log \left (x\right ) - 100 \, x\right )} e^{\left (\frac {2 \, e^{x}}{x}\right )} - 250 \, {\left ({\left (x - 1\right )} e^{x} \log \left (x\right )^{2} + {\left (x - 1\right )} e^{x} + 2 \, {\left ({\left (x - 1\right )} e^{x} - x\right )} \log \left (x\right ) - 2 \, x\right )} e^{\left (\frac {e^{x}}{x}\right )} + 625 \, {\left (2 \, {\left (x - 1\right )} e^{x} - x\right )} \log \left (x\right ) - 625 \, x\right )} e^{\left (-\frac {2 \, e^{x}}{x}\right )}}{x^{2}} \,d x } \]
integrate(((-200*x*log(x)+2*x^2-200*x)*exp(exp(x)/x)^2+((-500*x+500)*exp(x )*log(x)^2+((-1000*x+1000)*exp(x)+1000*x)*log(x)+(-500*x+500)*exp(x)+1000* x)*exp(exp(x)/x)+(1250*x-1250)*exp(x)*log(x)^2+((2500*x-2500)*exp(x)-1250* x)*log(x)+(1250*x-1250)*exp(x)-1250*x)/x^2/exp(exp(x)/x)^2,x, algorithm=\
integrate(2*(625*(x - 1)*e^x*log(x)^2 + 625*(x - 1)*e^x + (x^2 - 100*x*log (x) - 100*x)*e^(2*e^x/x) - 250*((x - 1)*e^x*log(x)^2 + (x - 1)*e^x + 2*((x - 1)*e^x - x)*log(x) - 2*x)*e^(e^x/x) + 625*(2*(x - 1)*e^x - x)*log(x) - 625*x)*e^(-2*e^x/x)/x^2, x)
Timed out. \[ \int \frac {e^{-\frac {2 e^x}{x}} \left (-1250 x+e^x (-1250+1250 x)+\left (-1250 x+e^x (-2500+2500 x)\right ) \log (x)+e^x (-1250+1250 x) \log ^2(x)+e^{\frac {2 e^x}{x}} \left (-200 x+2 x^2-200 x \log (x)\right )+e^{\frac {e^x}{x}} \left (e^x (500-500 x)+1000 x+\left (e^x (1000-1000 x)+1000 x\right ) \log (x)+e^x (500-500 x) \log ^2(x)\right )\right )}{x^2} \, dx=\int -\frac {{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^x}{x}}\,\left (1250\,x+{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^x}{x}}\,\left (200\,x+200\,x\,\ln \left (x\right )-2\,x^2\right )-{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x}}\,\left (-{\mathrm {e}}^x\,\left (500\,x-500\right )\,{\ln \left (x\right )}^2+\left (1000\,x-{\mathrm {e}}^x\,\left (1000\,x-1000\right )\right )\,\ln \left (x\right )+1000\,x-{\mathrm {e}}^x\,\left (500\,x-500\right )\right )-{\mathrm {e}}^x\,\left (1250\,x-1250\right )+\ln \left (x\right )\,\left (1250\,x-{\mathrm {e}}^x\,\left (2500\,x-2500\right )\right )-{\mathrm {e}}^x\,{\ln \left (x\right )}^2\,\left (1250\,x-1250\right )\right )}{x^2} \,d x \]
int(-(exp(-(2*exp(x))/x)*(1250*x + exp((2*exp(x))/x)*(200*x + 200*x*log(x) - 2*x^2) - exp(exp(x)/x)*(1000*x - exp(x)*(500*x - 500) + log(x)*(1000*x - exp(x)*(1000*x - 1000)) - exp(x)*log(x)^2*(500*x - 500)) - exp(x)*(1250* x - 1250) + log(x)*(1250*x - exp(x)*(2500*x - 2500)) - exp(x)*log(x)^2*(12 50*x - 1250)))/x^2,x)
int(-(exp(-(2*exp(x))/x)*(1250*x + exp((2*exp(x))/x)*(200*x + 200*x*log(x) - 2*x^2) - exp(exp(x)/x)*(1000*x - exp(x)*(500*x - 500) + log(x)*(1000*x - exp(x)*(1000*x - 1000)) - exp(x)*log(x)^2*(500*x - 500)) - exp(x)*(1250* x - 1250) + log(x)*(1250*x - exp(x)*(2500*x - 2500)) - exp(x)*log(x)^2*(12 50*x - 1250)))/x^2, x)